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Iterated privation and positive predication

Bjørn Jespersen a, Massimiliano Carrara b, Marie Duží a

aVSB-Technical University of Ostrava, Department of Computer Science, 17. listopadu 15, 708 33

Ostrava, Czech Republic

bFISPPA Department, Section of Philosophy, University of Padua, P.zza Capitaniato 3, 35139 Padova,

Italy

a r t i c l e i n f o a b s t r a c t

Article history:

Available online xxxx

Keywords:

Iterated modiﬁcation

Privative modiﬁcation

Property negation

Contraries

Requisite property

Intensional essentialism

Transparent Intensional Logic

TIL

The standard rule of single privative modiﬁcation replaces privative modiﬁers by

Boolean negation. This rule is valid, for sure, but also simplistic. If an individual

ainstantiates the privatively modiﬁed property (M F ) then it is true that a

instantiates the property of not being an F, but the rule fails to express the

fact that the properties (M F ) and Fhave something in common. We replace

Boolean negation by property negation, enabling us to operate on contrary rather

than contradictory properties. To this end, we apply our theory of intensional

essentialism, which operates on properties (intensions) rather than their extensions.

We argue that each property Fis necessarily associated with an essence, which is

the set of the so-called requisites of Fthat jointly deﬁne F. Privation deprives Fof

some but not all of its requisites, replacing them by their contradictories. We show

that properties formed from iterated privatives, such as being an imaginary fake

banknote, give rise to a trifurcation of cases between returning to the original root

property or to a property contrary to it or being semantically undecidable for want

of further information. In order to determine which of the three forks the bearers

of particular instances of multiply modiﬁed properties land upon we must examine

the requisites, both of unmodiﬁed and modiﬁed properties. Requisites underpin our

presuppositional theory of positive predication. Whereas privation is about being

deprived of certain properties, the assignment of requisites to properties makes

positive predication possible, which is the predication of properties the bearers must

have because they have a certain property formed by means of privation.

© 2017 Elsevier B.V. All rights reserved.

1. Introduction

There are large amounts of natural-language text data that need to be analyzed and formalized, because

we want to build up question-answering systems over these data. We want not only to convey information

explicitly recorded in these texts but also to derive implicit information entailed by these explicit data so

as to answer questions in an intelligent way. In other words, we want to apply logical reasoning to these

E-mail addresses: bjorn.jespersen@gmail.com (B. Jespersen), massimiliano.carrara@unipd.it (M. Carrara), marie.duzi@vsb.cz

(M. Duží).

https://doi.org/10.1016/j.jal.2017.12.004

1570-8683/© 2017 Elsevier B.V. All rights reserved.

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natural-language corpuses. To this end, we must analyze natural-language sentences in a ﬁne-grained way.

Since adjectives that denote property modiﬁers are part and parcel of our everyday vernaculars as well

as artiﬁcial languages, we need to logically analyze property modiﬁers as well. Privation being the most

complicated kind of property modiﬁcation, the goal of this paper is a ﬁne-grained analysis of privation

accompanied by rules governing reasoning about sentences that contain such modiﬁers.

Privative modiﬁcation is an operation that forms negated properties from properties. It is one among

three kinds of negation:

•privation, which applies to properties

• the complement function, which applies to sets

• the Boolean not, which applies to propositions-in-extension, i.e. truth-values.

When propositions are identiﬁed with (or at least modelled as) sets, then the complement function

subsumes propositional negation as a special case. Nothing in this paper hinges on this. What matters is

the contrast between privation, which is property negation and therefore an operation on intensions, and

set-theoretic negation, which takes a set to its complement and is therefore an operation on extensions.

The standard theory of modiﬁers is Montague Grammar, which is a typed version of model-theoretic

intensional logic. This paper provides an extension of this framework such that it is now possible to analyze

a particular sort of properties (or predicates, in the formal mode) that would previously fall outside the

purview of the framework. The paper also oﬀers reasons for revising one of the existing rules; however, the

extension we provide can be incorporated without revising anything. We are building upon the work of not

least Coulson and Fauconnier [3], Horn [13–15], Iwańska [16], Jespersen [17], Kamp [22], Montague [25],

Partee [27], Primiero and Jespersen [28], while the background theory is based on Duží et al. [6,8].

Montague Grammar comes with a well-entrenched logic for single privation. This framework states its

logic for the various modiﬁers in the form of elimination rules. The rule of single privation amounts to

replacing the privative modiﬁer by Boolean negation:

ais a fake banknote

ais not a banknote single privation

This rule is valid, because all that is required for validity is that the property (here, banknote) modiﬁed

by the privative modiﬁer (here, fake) not be predicated of a, and the conclusion achieves at least this much.

However, the above rule misses the internal link between the property of being a banknote and the property

of being a fake banknote. We will probe further into this point below, but the basic idea is that a fake

banknote is not just some object or other that fails to be a banknote, but rather it is an object that must

have a host of properties in common with banknotes. Though both forged banknotes and, say, weathercocks

and zebras are not banknotes, there is an intuitive sense in which forged banknotes are somehow ‘closer to’

banknotes than are weathercocks and zebras. The challenge before us is to deﬁne privation in such a way

that it is made explicit what banknotes and forged banknotes have, and must have, in common.

Another problem with the rule of single privation is that it fails to extend to iterated privation, as

Boolean negation can replace a privative modiﬁer only once. Here are some examples of predicates that

express iterated privation:

• ‘is an imaginary fake banknote’

• ‘is a former heir apparent’

• ‘is a former fallen angel’

• ‘weighs almost half a kilo’

• ‘is anything but a false friend’

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• ‘is a theory of non-antisymmetric mereology’

• ‘is an imaginary, burned fake banknote’

For instance, Horn in [13, pp. 296–308], [14,15] ponders the logic and rhetoric of double negatives, e.g.

as expressed by ‘not un-F’ (‘not unhappy’, ‘not impolite’, etc.).1Is a not impolite remark a polite remark,

perhaps even a very polite remark, as per litotes (cf. [14, pp. 86ﬀ]); or a remark that is neither polite nor

impolite, ending up in the neutral mid-interval? For a further example, consider so-called superdollars, which

are not US dollars, but counterfeit 100-dollar bills manufactured in, e.g., North Korea that are materially

(though not conceptually) well-nigh indiscernible from their genuine originals.2This particular occurrence

of ‘super’ in ‘superdollar’ has a privative eﬀect, so the predicate ‘is a fake superdollar’ expresses double

privation.3A fake superdollar is a fraudulent imitation of what is already a fraudulent imitation. If somebody

collects ﬁrst-degree counterfeit banknotes then they want a superdollar, and not a fake superdollar, which

exempliﬁes second-degree forgery by being a fake fake.4We all know that faking a fake will not return us

to the genuine original; but how do we know that? There is also the opposite direction: although you start

out with a 100-dollar bill, successfully passing it oﬀ as a fake superdollar to a collector of forged banknotes

of any degree, your 100-dollar bill has not transmogriﬁed into a fake superdollar, despite being accepted as

one. But how do we know that? The answer we will be pursuing is that we know that because we know the

meaning of the respective predicates.

But what to replace Boolean negation with in order to develop a logic of iterated privation? We suggest

replacing Boolean negation with property negation. First, property negation operates on properties, just as

property modiﬁers do, so the intensional character of modiﬁcation is carried through to negation. Second,

property negation obeys a logic of contraries rather than contradictories, which provides the kind of rule

that privative modiﬁcation requires.

Let us take a closer look at privation. There are two material sources of privation. One is resultative and

hence diachronic: individual aonce was an F, but is no longer an F.5A recaptured fugitive (cf. [11]) once

was a fugitive, but is no longer one. Finished meals, burnt (not just charred) pieces of meat, and obsolete

banknotes all exemplify resultative privation. Given the actual laws of nature, neither a ﬁnished meal, nor

a burnt piece of meat can again become a meal or a piece of meat, whereas an obsolete banknote might be

restored to its previous glory as a banknote should the social institutions so favour it. The other source is

achronic: adid not start out as an Fand might never become an F, although it is possible that amight in

fact become an F, as when the relevant social institutions decree that such-and-such fake banknotes shall

henceforth acquire the status of valid tender, thus turning them into banknotes. Only this latter property

is extraneous to the property of being a fake banknote.

There are two formal sources of privation: either by way of ﬁrst-degree or higher-degree modiﬁcation.

Either a privative modiﬁer modiﬁes a property that has already been modiﬁed by a privative modiﬁer, as

when imaginary is applied to fake banknote. Or a privative modiﬁer modiﬁes another privative modiﬁer,

and the resulting modiﬁer is applied to a property, as when anything but is applied to false and the resulting

modiﬁer, anything but false, is applied to friend.6(In this paper we shall consider only ﬁrst-degree iterated

1See Horn [13, pp. 38–41] for a historical survey of various takes on contrariety and predicate term negation.

2Cf. http://en.wikipedia.org/wiki/Superdollar.

3It is not an open-and-shut issue whether some modiﬁers are absolutely privative while the rest are context-sensitive by being

privative only with respect to some argument properties. Fake might be an example of the former, though we are issuing no

guarantee. Examples of the latter would include Nordic gold, which is not gold (but an alloy); ﬁdes punica, which is not trust (but

treachery); a baker’s dozen, which is not a dozen (but thirteen); and Rocky Mountain oysters, which are not oysters. See also [16]

on context-sensitive privatives.

4We could shift both the real McCoys and the fakes one level up with collectors collecting second-degree fakes and being fooled

by third-degree fakes; and so on up.

5Other dynamic examples of ‘stages of loss in the privative process’ and ‘incomplete realizations of possible privational histories’

(Martin [24, p. 439, 441, resp.]) would include going bald, i.e. progressing (or perhaps regressing) toward being almost or entirely

without hair.

6Anything but is a privative intensiﬁer, just like very is a subsective intensiﬁer, as in very good.

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privation in the interest of brevity.) What we just described is double privation, but the theory readily

generalizes to triple, quadruple (etc.) privation, as when imaginary is applied to burned fake banknote.

In the light of the fact that privative modiﬁers can be nested, one may wonder: could we avail ourselves

of a rule that would calculate, for an arbitrary string of privative modiﬁers of two or more, whether the

root property Fis true of a? Our research shows that no such rule is forthcoming. There can be no unique

rough-and-ready rule for iterated privation.

Iterated privation issues instead in a trifurcation of cases:

(i) ais an F

(ii) afails to be an F

(iii) it is semantically indeterminate whether ais an F

The fact that this trifurcation emerges reﬂects the nature of privation. The ﬁrst of two general points

bearing on privation is the negative one that privation is about what something is not, or fails to be. It

is about one or more properties that an object is deprived of. In particular, no theory of privatives should

predict that fake banknotes are extracted from sets of banknotes: fake banknotes are not banknotes that

are fake.7

But the second point is the positive one that there is substantially more to privation than deprivation.

Let Fbe a property, Mpa modiﬁer privative with respect to F, and [MpF] the privatively modiﬁed property

that results from applying the modiﬁer to the root property. The intuition we wish to capture is that when

an object has the property [MpF] then the object is—in some sense yet to be made clear—‘closer’ to having

Fthan are many or most other objects that lack the privatively modiﬁed property [MpF].8By way of an

example, a fake banknote is ‘almost’ a banknote, deﬁnitely barred from being one, yet it has a greater overlap

in terms of properties with a banknote than have most other objects. Fake banknotes must share a host

of properties with banknotes; otherwise they could not be fake banknotes in the ﬁrst place, but would be

merely, say, colourful slips of paper. For instance, a banknote must mention the issuing authority, a currency

and a denomination. Therefore, a fake banknote must also mention an issuing authority, a currency and a

denomination. If a fake banknote sports, for instance, the words ‘ECB’, ‘EURO’, ‘100’ and is printed on

cotton-based paper with the look and feel of garden-variety banknotes then it lends itself to several instances

of what we call positive predication. Positive predication predicates properties of an object which the object

must instantiate and which are not privatively modiﬁed.

Positive predication appears to be less complicated vis-à-vis achronic privation than diachronic privation.

A burnt piece of meat is ash (inorganic matter) and in this second state not at all close to being meat (organic

matter), whereas a fake banknote must be close to being a banknote. However, we are able to put forward

a theory of positive predication with regard to objects that exemplify privatively modiﬁed properties of

either kind, because we oﬀer a presuppositional theory of privation. The theory is presuppositional because,

for an object to exemplify a privatively modiﬁed property, it is presupposed that the object should already

exemplify other properties. By way of illustration, the property of being a former smoker comes with the

presupposition that, as a matter of analytic necessity, whoever currently instantiates it previously, but no

longer, instantiated the property of being a smoker. Or if ahas the property of being a Vatican cardinal

then a has also the property of being ﬂuent in Latin. Beyond the well-rehearsed example of former smokers,

the theory extends to not only social artefacts like positions in a hierarchy of institutional power, but also

7This is to say that we do not stretch the meaning of ‘is a banknote’ so as to include fake banknotes among the banknotes. Partee

[27] suggests using coercion to do just that, such that it becomes meaningful to inquire whether a banknote is a real banknote or

a fake banknote. Jespersen [17, p. 544, fn. 14] and Duží et al. [6, p. 400, fn. 52] argue against Partee’s suggestion.

8Coulson and Fauconnier [3] and Iwańska [16] also think of privation both as the elimination of some, but not all, properties

(or concepts, features, etc.) inhering in privatively modiﬁed properties, and as the ‘blending with’ or ‘introduction of’ additional

properties so as to form new, hybrid properties like stone lion or toy elephant.

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technical artefacts like tools and scientiﬁc frameworks like taxonomies.9The presuppositional theory enables

us to infer conclusions about, say, dead whales (a dead whale being a dead mammal), disassembled watches

(a disassembled watch not being a timekeeping device) and burnt pieces of meat (a burnt piece of meat

having been previously a piece of meat). Importantly, each and every property we countenance has a host

of other properties associated with it. Thus, each instance of [Mp[MpF]] also comes with a host of adjacent

properties that their respective bearers must also bear.

We call the adjacent properties requisites.10 Our thesis is that privation is the deprivation of some, but

not all, requisites. The surviving requisites, together with some added ones, form the basis of positive

predication. The above trifurcation arises because the root property Fwill be one of the requisites of

some multiply privatively modiﬁed properties, while non-Fwill be one of the requisites of other multiply

privatively modiﬁed properties, whereas neither F, nor non-Fis among the requisites of still other multiply

privatively modiﬁed properties. In the third and ﬁnal case, as far as the semantics of such properties goes,

there is no semantic fact of the matter as to which side of the fence acomes down on. Extra-semantic,

empirical investigation must, in each individual case, determine which side a given individual comes down

on. To give a taste of the trifurcation, here is an example of each of its three horns.

• If somebody is anything but a false friend then they are a friend (and that to a very high degree) (i).

• If something weighs almost half a kilo then it weighs less than a kilo and, therefore, does not weigh a

kilo (ii).11

• If someone is a former heir apparent then either they are now the incumbent monarch or they are no

longer being even considered for the throne (iii).

The requisites of a given property enable valid reasoning from assumptions about privatively modiﬁed

properties. Philosophically speaking, associating requisites with properties amounts, in the case of privation,

to laying down at least some of what goes into being a wooden horse, a burnt wooden horse, a burnt fake

wooden horse, a fake burnt wooden horse, etc. Achieving the latter, philosophical, objective comes with a

fair amount of idealization while still requiring substantial philosophical justiﬁcation.12 In this paper we rest

content with setting out the formal features of the framework within which we discuss iterated privation

and positive predication. Just to be clear, while we will be arguing for a particular elimination rule for

privatives, we will not attempt to put forward any introduction rules for privatives in the vein of:

P1,...,Pn

ais an [MpF]

For particular instances of Piand [MpF], such a rule would make explicit what the conditions are for

being, e.g., a fake banknote, or a wooden horse, or a malfunctioning toothbrush.13 Philosophy of technology

would make a great leap forward if particular instances of Picould be spelt out with the rigour required

9For the relevant theory of presuppositions, see [7,9,8].

10 The notion of requisite was conceived by Tichý and introduced in [30, p. 408]. It has subsequently been turned into a theory of

intensional essentialism. See Jespersen and Materna [21], Duží et al. [6, Ch. 3].

11 We are making the fairly uncontroversial assumption that when something weighs almost half a kilo then it weighs no more

than that. We want to blot out the kind of scenario where something that weighs exactly a kilo weighs also 900 grams, almost half

a kilo, etc., in virtue of a simple argument of downward monotonicity that also validates the ‘countdown’ inference that if you have

ﬁve ﬁngers on your hand then you also have four (three, ..., zero) ﬁngers, which still does not entail that you have ﬁfteen ﬁngers.

12 See [2].

13 It is not a matter of course that malfunctioning is privative. It is on the causal-role theory of technical function (what cannot

hammer cannot be a hammer), whereas it is subsective on the proper-function theory (a malfunctioning hammer was still designed

to hammer as its proper function). See Jespersen and Carrara [20]. An interesting study on malfunctioning software has been

recently provided by Floridi, Fresco and Primiero [12]. The authors distinguish between two kinds of malfunctioning software,

namely in terms of ‘negative’ dysfunction and ‘positive’ misfunction. They argue that while dysfunction is the core property of

malfunctioning technical artefacts, an executed software token cannot dysfunction, because it will always work in accordance with

its design. Yet it can, and often will, misfunction, because the design does not completely live up to the intended speciﬁcation.

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for an introduction rule. But, although the notion of requisite would come in handy, this very ambitious

enterprise is beyond the compass of this paper.14

The fundamental distinction among modiﬁers is typically considered to be one between the subsectives

and the non-subsectives.15 The former group would consist of the pure subsectives (that are governed by the

upwardly monotonic rule of right subsectivity, which amounts to eliminating the modiﬁer and predicating

the surviving property) and the intersectives (that are governed by the rule of right subsectivity and a rule

of left subsectivity).16

Here is a brief comparison in prose of the four standard types of modiﬁers, where an index is an index of

empirical evaluation, such as a possible world or a world/time pair.

•Pure subsectives. At every index a skilful surgeon is a surgeon.

•Intersectives. At every index a round peg is round and is a peg.

•Privatives. At no index is a fake banknote a banknote.

•Modals. At some indices an alleged assassin is an assassin, and at some other indices an alleged assassin

is not an assassin.

Montague [25, p. 211] seeks to provide a uniform theory of modiﬁers (strictly speaking, of adjectival

phrases). Each modiﬁer, according to Montague, is a property-to-property mapping.17 We subscribe to

this uniform account of the corpus of modiﬁers. We depart, however, from Montague’s contention that

these functions are meaning-to-meaning functions (ibid.). The contention, of course, makes perfect in Mon-

tague’s intensional framework in which intensions (functions whose domain are the logical space of possible

worlds) count as meanings.18 In our framework, meaning-to-meaning functions would be hyperintension-to-

hyperintension functions. We have such functions, but we do not need them here. We do need hyperinten-

sions, however, when working with modiﬁers: we need hyperintensions (meanings) when deﬁning a couple

of key notions that go into deﬁning modiﬁers. In a word, we are using hyperintensions in order to operate

on intensions.

It is relevant to compare modal and iterated privative modiﬁcation, for in neither case is only one

conclusion possible. The modals require extra-semantic, empirical inquiry to establish, for each particular

instance, which of two ways the facts happen to go. Only empirical inquiry can decide which allegations of

being an assassin are true and which ones are false. The iterated privatives require intra-semantic inquiry

to establish, for each particular instance, which of the three ways the meanings go. If we land on the third

fork, then we need to get out of the armchair and into the ﬁeld to establish which way the facts happen to

go.

Privation is literally radical modiﬁcation, because the root property is modiﬁed away. Subsective modi-

ﬁcation, by contrast, enriches the root property, whether the modiﬁer be intersective (e.g. round ) or purely

subsective (e.g. skilful). A peg, say, is qualiﬁed as a round peg, or conversely, something round is qualiﬁed

as a round peg; and a surgeon as a skilful surgeon. A layer of modiﬁcation is added on top of the existing

requisites of the root property. Privation goes in the opposite direction by purging the root property of some

of its requisites. This is the crucial step toward explaining why a fake banknote fails to be a banknote. One

property that drops out is that of being valid tender, which comes with requisites of its own. Yet privation

14 See Del Frate [5] for conceptual discussion of a catalogue of engineering conceptions of malfunction.

15 See, e.g., Makinson [23, pp. 64–65] on the distinction between qualiﬁers and proper modiﬁers.

16 See Jespersen [17] for two rules of left subsectivity.

17 A topic we will not be delving further into here is how to decide for a given token of a given adjective whether it denotes a

property or a modiﬁer. See, however, Siegel [29], Kamp [22], Montague [25], Beesley [1]. Schematically put, Montague pairs all

adjectives oﬀ with modiﬁers, Beesley pairs all adjectives oﬀ with properties, and Kamp pairs some adjectives oﬀ with modiﬁers

and the rest with properties.

18 We are glossing over the facts that Montague did not fully commit to s(i.e. combined world/time pairs) as a stand-alone type

on an equal footing with e(i.e. ‘entity’), t(i.e. truth-value), etc., and that Montague’s empirical indices were combined world/time

pairs.

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not only detracts, but also adds something. One property that gets added is that of being a forgery (i.e. a

fraudulent imitation of something or other), which also comes with requisites of its own. The crucial step

toward the presuppositional theory required for positive predication is that privation adds new requisites to

the purged set of requisites. Moreover, some of these new requisites contradict some of the original purged

requisites. This explains why we can predicate several properties of fake banknotes that they must have.

If we did not assign requisites to properties, we would be left with an exceptionally minimalist logic of

iterated privation. First of all, the replacement of privatives by Boolean negation can occur only once, as

we announced at the outset. Here is why. The standard rule of single privation lays down what to do when

it is true that ahas property [MpF]. The rule fails to state what to do when the premise is the negation

that ahas property [MpF]. This inference, therefore, is invalid:

¬[[MpF]a]

¬¬[F a]

If, counterfactually, the rule for privation had speciﬁed logical equivalence between [[MpF]a] and ¬[F a]

then the above argument would have come out valid. However, the rule of privation does not specify

equivalence; rather it speciﬁes that [[MpF]a] entails ¬[F a]. It is also intuitive enough that the above

inference must come out invalid. If it did not, all instances of double privation would land on the ﬁrst fork.

Thus, a fake rhinestone diamond would emerge as a diamond. So not only would the inference fail to be

truth-preserving by over-generating instances of the ﬁrst fork, it would also leave no room for the other two

forks.

Secondly, therefore, in the interest of setting up a logic of iterated privation, we suggest replacing Boolean

negation by property negation, denoted by ‘non’. This replaces contradictories by contraries, which makes

for a suﬃciently weakened form of negation. Applied to single privation, the result is:

ais a fake banknote

ais a non-banknote single privation∗

When ais a fake banknote at some index then ais sent to the complement set of the set of banknotes

at the same index, though not to just anywhere in the complement, but to its particular subset of fake

banknotes. The good news is that we can reiterate non so as to form the property non-non-banknote. The

bad news is that [non [non F ]] would be the ﬁnal word on iterated privation in the absence of requisites.

The above trifurcation would remain, but it would be impossible to decide which particular fork a particular

instance of iterated privation landed on. A logic of iterated privation that amounted to replacing privatives

by non would grind to a halt after having established the general insight that pairs of privatives yield

contraries.

The thesis, then, that we are arguing for can be condensed thus. A logic of iterated privation that invokes

requisite properties of privatively modiﬁed properties enables positive predication and is in a position to

land particular instances of multiply privatively modiﬁed properties on the right fork.

The rest of the paper is organized as follows. Section 2 sets out the relevant portions of our formal

semantic theory. Section 3 compares the logic of subsectives against the logic of privatives, introduces

property negation, and oﬀers case studies of each of the forks of the trifurcation.

2. Logical foundations

In this section, we set out the formal framework within which we raise and solve the problem of iterated

privation. The framework is a fragment of Tichý’s Transparent Intensional Logic (TIL). The relevant frag-

ment is more or less continuous with Montague’s intensional logic and its myriad extensions. However, TIL

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has added a theory of modal modiﬁers (see Primiero and Jespersen [28]) to the Montagovian corpus, and

a spelt-out logic, including an introduction rule, for intersective modiﬁers (see Jespersen [17]), as well as a

general rule of left subsectivity applying to privatives and modals, intersectives and pure subsectives (see

Jespersen [17]; Duží et al. [6, §4.4]), which extends to single privation. The present paper is the third and

ﬁnal of a trilogy of papers on how to model various states (especially malfunction) of technical artefacts by

means of property modiﬁers. The two preceding papers are Jespersen and Carrara [19,20].

2.1. Key deﬁnitions of Transparent Intensional Logic

We need deﬁnitions of the following basic notions:

•Simple type theory. We need this deﬁnition in order to deﬁne both intensional and extensional entities.

Properties of individuals are typed as functions from possible worlds to functions from times to sets of

individuals, where sets are identiﬁed with their characteristic functions. Property modiﬁers are typed

as property-to-property functions. (Modiﬁer modiﬁers are typed as modiﬁer-to-modiﬁer functions.)

•Constructions. We need this deﬁnition for the following reasons. Constructions are (ﬁne-grained and

structured) meanings; we deﬁne four of the altogether six constructions that make up the full inductive

deﬁnition of constructions. Furthermore, the deﬁnition introduces the formalism of TIL, which is based

on λ-abstracts.

•Requisite. The requisite relation Req is a relation-in-extension between two properties Rand P, such

that, necessarily, whatever is (in) the extension of Pmust, as a matter of analytic necessity, also be (in)

the extension of R, though not necessarily conversely. We say that Ris a requisite of P.

•Essence. The essence of a property Pis the set of its requisites which together deﬁne P.

•Property negation. Property negation, non, allows iteration and obeys a logic of contraries.

Note that our theory is based on what we call intensional essentialism. The analytically necessary relation

of being a requisite of Pand being an element of the essence of Pobtains between intensional entities such as

properties, and not between extensional entities (such as individuals) and intensional entities. Consequently,

we subscribe to individual anti-essentialism: no individual has any purely contingent property necessarily.

By ‘purely contingent property’ we mean a non-constant property that does not have what we call an

essential core; e.g., the property of having exactly as many inhabitants as Prague is necessarily exempliﬁed

by Prague, whatever number of inhabitants Prague may happen to have.19

We deﬁne the essence of a property as a set of its requisites that jointly deﬁne the property. For instance,

the property of being a mammal is related by the requisite relation to the property of being a whale.

Thus, necessarily, if the individual ahappens to be a whale at a world/time index of evaluation then ais

also a mammal at this world/time. It is an open question (epistemologically and ontologically speaking)

whether ais a whale. Establishing whether it is one requires investigation a posteriori. On the other hand,

establishing whether amust be a mammal in case ahappens to be a whale is a priori, the requisite relation

being in-extension and as such independent of what is true at any particular state of aﬀairs. Comparing the

essences of a root property and a modiﬁed property enables us to deﬁne subsective and privative modiﬁers

in a new way that is an extension of previous deﬁnitions.

19 See Duží et al. [6, §1.4.2.1] for a classiﬁcation of empirical properties and (ibid.: 68) for the notion of essential core.

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Deﬁnition 1 (Simple type theory). Let Bbe a base, where a base is a collection of pair-wise disjoint, non-empty

sets. Then:

i) Every member of Bis an elementary type of order 1 over B.

ii) Let α,β1, ..., βm(m > 0) be types of order 1 over B. Then the collection (α β1...βm) of all m-ary partial

mappings from β1×... ×βminto αis a functional type of order 1 over B.

iii) Nothing is a type of order 1 over B unless it so follows from (i) and (ii). ✷

Notation. That an object Ois of type α, i.e. belongs to the type α, will be denoted ‘O:α’.

Remark. For the purposes of natural-language analysis TIL uses the following so-called objectual base B

consisting of the following atomic types:

o: the set of truth-values T, F;

ι: the set of individuals (the universe of discourse);

τ: the set of real numbers (doubling as discrete times);

ω: the set of logically possible worlds (the logical space).

Deﬁnition 2 (Constructions).

(i) Variables x,y, ... are constructions that construct objects (elements of their respective ranges) depen-

dently on a valuation v; they v-construct.

(ii) Where Xis an object whatsoever (an extension, an intension or a construction), 0

Xis the construction

Trivialization.0

Xconstructs Xwithout any change in X.

(iii) Let X,Y1, ..., Ynbe constructions. Then Composition [X Y1...Ym] is the following construction. If X

v-constructs a function fof type (αβ1...βm), and Y1, ..., Ymv-construct entities B1, ..., Bmof types β1,

..., βm, respectively, then [X Y1...Ym]v-constructs the value (an entity, if any, of type α) of fon the

tuple-argument hB1, ..., Bmi. Otherwise [X Y1...Ym] does not v-construct anything and so is v-improper.

(iv) The Closure [λx1...xmY] is the following construction. Let x1, x2, ..., xmbe pair-wise distinct vari-

ables v-constructing entities of types β1, ..., βm, respectively, and Ya construction typed to v-construct

an α-entity. Then [λx1...xmY] is the construction Closure (or λ-Closure). It v-constructs the follow-

ing function f: (αβ1. . . βm). Let v(B1/x1, ..., Bm/xm) be a valuation identical with vat least up

to assigning objects B1:β1, ..., Bm:βmto variables x1, ..., xm. If Yis v(B1/x1, ..., Bm/xm)-improper

(see iii), then fis undeﬁned on hB1, ..., Bmi. Otherwise the value of fon hB1, ..., Bmiis the α-entity

v(B1/x1, ..., Bm/xm)-constructed by Y.

(v) Nothing is a construction, unless it so follows from (i) through (iv). ✷

Remark. That a variable x v-constructs entities of a type αwill be referred to as ‘ranging over α’, denoted

by ‘x→vα’. We model sets and relations by their characteristic functions. Thus, for instance, (oι) is the

type of a set of individuals, while (oιι) is the type of binary relations-in-extension between individuals.

Empirical expressions denote empirical conditions that may or may not be satisﬁed at some world/time

pair of evaluation. We model these empirical conditions as possible-world-semantic (PWS) intensions. PWS

intensions are entities of type (βω): mappings from possible worlds to an arbitrary type β. The type β

is frequently the type of the chronology of α-objects, i.e., a mapping of type (ατ ). Thus α-intensions are

frequently functions of type (α(τω)), abbreviated as ‘ατ ω ’. Extensional entities are entities of the arbitrary

type αwhere α6= (βω) for any type β. Where wranges over ωand tover τ, the following logical form

essentially characterizes the logical syntax of empirical language: λwλt [...w...t...].

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Examples of frequently used PWS intensions are:

•propositions of type oτ ω

•properties of individuals of type (oι)τ ω

•binary relations-in-intension between individuals of type (oιι)τ ω

•individual oﬃces (or roles) of type ιτω

Logical objects like truth-functions and quantiﬁers are extensional: ∧(conjunction), ∨(disjunction), ⊃

(implication) are of type (ooo), and ¬(Boolean negation) of type (oo). Since TIL has no syncategorematic

symbols, all the symbols in the TIL formalism denote functions, including quantiﬁers. The quantiﬁers ∀α,

∃αare type-theoretically polymorphic total functions, just as in Montague Grammar, of type (o(oα)), for

an arbitrary type α, and are deﬁned as follows.

Deﬁnition 3 (Quantiﬁers). The universal quantiﬁer ∀αis a function of type (o(oα)) that takes a class A

of α-elements to Tif Acontains all elements of the type α, otherwise to F. The existential quantiﬁer ∃α

is a function of type (o(oα)) that takes a class Aof α-elements to Tif Ais a non-empty class, otherwise

to F.✷

Notational conventions.

• ‘∀x . . .’ serves as a shorthand for ‘[0∀λx . . .]’; similarly for ‘∃y’: all variable-binding is λ-binding, and

universal (existential) quantiﬁcation is presented by means of Trivialization.

• Below all type indications will be provided outside the formulae in order not to clutter the notation.

• The outermost brackets will be omitted whenever no confusion can arise.

• While ‘X:α’ means that an object Xis (a member) of type α, ‘X→vα’ means that Xis typed to

v-construct an object of type α, if any. We write ‘X→α’ if no confusion concerning valuation arises.

•w→vωand t→vτ.

• If C→vατ ω then the frequently used Composition [[C w]t], which is the intensional descent (a.k.a.

extensionalization) of the α-intension v-constructed by C, will be encoded as ‘Cwt ’.

Predication is an instance of Composition.20 An empirical predicate such as ‘is a planet’ denotes the

property of being a planet; it is subsequently extensionalized in order to obtain the set of planets at the

empirical indices of evaluation; the characteristic function of the set is applied, by way of Composition, to

the individual of which the property of being a planet is predicated; the result (a truth-value) is ﬁnally

abstracted over by means of wand tvariables in order to construct an empirical truth-condition of type

oτ ω. The form of the predication of being a planet of an individual ais this:21

λwλt [0

P lanetwt 0

a]

The form of the predication of the subsectively modiﬁed property of being a gas planet is this:

λwλt [[0Gas 0

P lanet]wt 0

a]

To begin, construct, by way of Composition, the property of being a gas planet and then follow the same

steps as above.

20 See Duží et al. [6, §2.4.2] on predication.

21 We apply the method of analysis according to which semantically simple predicates like ‘is a planet’ are associated with the

Trivialization of the denoted object; 0P lanet, in this case. See Duží et al. [6, §2.1].

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Types: Planet : (oι)τ ω;a:ι;Gas : (((oι)τ ω)((oι)τ ω )); [0Gas 0

P lanet]→(oι)τ ω ; [[0Gas 0

P lanet]wt 0a]→vo;

λwλt [[0

Gas 0

P lanet]wt 0

a]→oτ ω: the proposition that ais a gas planet.

2.2. Requisites

The requisite relations Req are a family of relations-in-extension between two intensions, so they are of

the polymorphous type (o ατ ω βτ ω ), with the possibility that α=β.22 Inﬁnitely many combinations of Req

are possible, but for our purpose we will need just this one:

Req : (o(oι)τ ω (oι)τ ω )

Req is a relation between two properties of individuals, such that one is a requisite of the other.

TIL embraces partial functions.23 Partiality gives rise to the following complication. The requisite relation

obtains necessarily, i.e. for all worlds wand times t, and so the values at this or that hw,tiof particular

intensions are irrelevant. But the extensions of properties (i.e. sets) are isomorphic to characteristic functions,

and these functions are amenable to truth-value gaps. As already mentioned, the property of having stopped

smoking comes with a bulk of requisites including not least the property of being a former smoker. Thus,

the predication of such a property Pof amay also fail, causing [0

Pwt 0

a] to be v-improper. There is a

straightforward remedy, however, namely the propositional property of being true at hw,ti;True: (o oτ ω )τ ω .

Given a proposition Prop, [0T ruewt 0

P rop]v-constructs Tif Prop is true at hw,ti; otherwise (i.e., if Prop is

false or else undeﬁned at hw,ti) it v-constructs F.

Deﬁnition 4 (Requisite relation between ι-properties). Let X,Ybe constructions such that X,Y→(o ι)τ ω ;

x→ι. Then

[0

Req Y X] = ∀w∀t[∀x[[0T ruew t λwλt [Xwt x]] ⊃[0T r uewt λwλt [Ywt x]]]].

Gloss deﬁniendum as, “Yis a requisite of X”, and deﬁniens as, “Necessarily, i.e. at every hw,ti, any x

that instantiates Xat hw,tialso instantiates Yat hw,ti.”

Example. Let the property of being a person be a requisite of the property of being a student. Then the

hyperproposition that all students are persons is an analytic truth. It constructs the proposition TRUE,

which is the necessary proposition, which takes value Tat all world/time pairs. Wherever and whenever

somebody is a student they are also a person. Formally:

[0

Req 0

P erson 0

Student] = ∀w∀t[∀x[[0T r uewt λwλt [0

Studentwt x]] ⊃[0T ruewt λwλt [0

P ersonwt x]]]]

Claim 1. Req is a quasi-order on the set of ι-properties.

Proof. Let X,Y→(oι)τ ω . Then Req belongs to the class QO:(o(o(oι)τ ω (oι)τ ω)) of quasi-orders over the

set of individual properties:

22 For comparison, Jespersen [18] oﬀers a detailed study of a requisite relation, of type (o ιτω ιτω ), where one individual oﬃce is a

requisite of another individual oﬃce, the way the oﬃce of Commander-in-Chief is a requisite of the oﬃce of President of the United

States. The paper analyses “Superman is Clark Kent” as expressing that this particular requisite relation obtains between one

oﬃce denoted by ‘Superman’ and another oﬃce denoted by ‘Clark Kent’. If you occupy the oﬃce of Superman you must co-occupy

the oﬃce of Clark Kent, but you can occupy the Clark Kent oﬃce without occupying the Superman oﬃce. This goes to show

that TIL oﬀers an intensional analysis (based on intensional essentialism) of “Superman is Clark Kent”, contrary to the prevalent

‘Millian’ extensional analyses.

23 See Duží et al. [6, 276–78] for a philosophical justiﬁcation of partiality in spite of the associated technical complications.

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Reﬂexivity. [0

Req X X ] = ∀w∀t[∀x[[0T ruewt λwλt [Xwt x]] ⊃[0T ruewt λw λt [Xwt x]]]].

Transitivity. We want to prove that [[[0

Req Y X]∧[0

Req Z Y ]] ⊃[0

Req Z X]].

1. [[0

Req Y X]∧[0

Req Z Y ]] ∅

2. [∀w∀t[∀x[[0T ruewt λwλt [Xwt x]] ⊃[0T ruewt λw λt [Ywt x]]]] ∧

∀w∀t[∀x[[0T ruewt λwλt [Ywt x]] ⊃[0T ruewt λw λt [Zwt x]]]]] 1, Deﬁnition 4

3. [[0T ruewt λwλt [Xwt x]] ⊃[0T ruewt λwλt [Ywt x]]] 2, ∀E,∧E

4. [[0T ruewt λwλt [Ywt x]] ⊃[0T ruewt λw λt [Zwt x]]] 2, ∀E,∧E

5. [[0T ruewt λwλt [Xwt x]] ⊃[0T ruewt λwλt [Zwt x]]] 3, 4, *

6. [∀w∀t[∀x[[0T ruewt λwλt [Xwt x]] ⊃[0T ruewt λw λt [Zwt x]]]] 5, ∀I

7. [[0

Req Z X] 6, Deﬁnition 4

8. [[[0

Req Y X]∧[0

Req Z Y ]] ⊃[0

Req Z X]] 7, ⊃I

Remark. In line (5) ‘*’ denotes the theorem of the transitivity of implication.

In order for a requisite relation to be a weak partial order, it would need to be also anti-symmetric. The

Req relation is, however, not anti-symmetric. If properties X,Yare mutually in the Req relation, i.e. if

[[0

Req Y X]∧[0

Req X Y ]]

then at every hw, tithe two properties are true of exactly the same individuals. This does not entail, however,

that X,Yare identical. It may be the case that there is an individual asuch that [Xwt a]v-constructs F

whereas [Ywt a] is v-improper. For instance, the following properties X,Ydiﬀer only in truth-value for those

individuals who never smoked. Let StopSmoke:(oι)τω be the property of having stopped smoking. Whereas

Xyields truth-value gaps on such individuals, Yis false of them:

X=λwλt λx [0

StopSmokew t x]

Y=λwλt λx [0T ruewt λw λt [0

StopSmokew t x]]

This makes for a negligible diﬀerence that can be abstracted away, so we introduce the equiva-

lence relation ≈: (o(oι)τ ω(oι)τ ω ) on the set of individual properties; p, q →(oι)τ ω; = : (ooo); =df :

(o(o(oι)τ ω(oι)τ ω )(o(oι)τ ω (oι)τ ω )), i.e. the identity of binary relations between properties.

0≈=λpq [∀x[[0T ruewt λwλt [pwt x]] = [0T ruewt λw λt [qwt x]]]]

Now we can deﬁne the Req’ relation on the factor set of the set of ι-properties as follows.24

Let [p]≈=λq [0≈p q] and [Req′[p]≈[q]≈] = [Req p q ]. Then:

Claim 2. Req’ is a partial order on the factor set of the set of ι-properties with respect to the relation ≈.

Proof. It is suﬃcient to prove that Req’ is well-deﬁned. Let p,qbe ι-properties such that [ 0≈p p′] and

[0≈q q′]. Then:

[Req′[p]≈[q]≈] = [Req p q]

=∀w∀t[∀x[[0T ruewt λwλt [pwt x]] ⊃[0T ruewt λw λt [qwt x]]]]

=∀w∀t[∀x[[0T ruewt λwλt [p′

wt x]] ⊃[0T ruewt λwλt [q′

wt x]]]] = [Req′[p′]≈[q′]≈]

24 The deﬁnition of Req’ was ﬁrst introduced in Duží et al. [6, 363–364].

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Now, obviously, the relation Req’ is anti-symmetric:

[[0

Req′[p]≈[q]≈]∧[0

Req′[q]≈[p]≈]] ⊃[[p]≈= [q]≈]

To make the exposition easier to follow, in what follows we will neglect this minor diﬀerence between

properties λwλt λx [0T r uewt λwλt [pwt x]] and pso that instead of the former we will write simply ‘p’. ✷

2.3. Intensional essentialism

Next, we are going to deﬁne the essence of a property. Our essentialism is based on the idea that since no

purely contingent property can be essential of any individual, essences are borne by intensions rather than

by individuals exemplifying intensions. That a property Fhas an essence means that a relation-in-extension

obtains a priori between Fand a set Essence of other properties such that, as a matter of analytic necessity,

whenever an individual (an ι-entity) instantiates Fat some hw , tithen the same individual also instantiates

all the properties belonging to Essence at the same hw, ti. Hence our essentialism is based on the requisite

relation, couching essentialism in terms of a priori interplay between properties, regardless of who or what

exempliﬁes a given property. The essence of a property Fis identical to the set of requisites that jointly

deﬁne F. The hw, ti-relative extensions of a given property are irrelevant, as we said; but so are the various

equivalent constructions of the property.

Deﬁnition 5 (Essence of a property). Let p, q →(oι)τ ω ;Ess: ((o(oι)τ ω )(oι)τ ω ), i.e. a function assigning to a

given property pthe set of its requisites deﬁned as follows:

0

Ess =λpλq [0

Req q p]

Then the essence of a property pis the set of its requisites:

[0

Ess p] = λq [0

Req q p]

Each property has many requisites. The question is: how do we know which properties are the requisites

of a given property? The answer requires an analytic deﬁnition of the given property. For instance, consider

the property of being a (domestic) cat. A classiﬁcation according to biological taxonomy can serve as such

a deﬁnition:

Kingdom: Animalia

Phylum: Chordata

Clade: Synapsia

Class: Mammalia

Order: Carnivora

Family: Felidae

Subfamily: Felinae

Genus: Felis

Species: Felis Catus

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Thus, we can deﬁne a cat as an animal belonging to all of the above categories.25 From this deﬁnition it

follows that, for instance, the sentence “Cats are mammals” comes out analytically true:

∀w∀t[∀x[[0

Catw t x]⊃[0

Mammalwt x]]]

Hence the property of being a mammal is a requisite of the property of being a cat. All the above properties

deﬁned by a given taxonomy belong to the essence of the property of being a cat.

3. Subsectives, privatives, property negation, and case studies

3.1. Subsectives and privatives

With the above deﬁnitions in place, we can go on to compare two kinds of subsectives against privatives:26

• A modiﬁer Mis non-trivially subsective with respect to property Fiﬀ the modiﬁed property [M F ] has

all the requisites of Fand at least one additional requisite that is not a requisite of F. In other words,

the essence of Fis a proper subset of the essence of [M F ].

For instance, a skilful surgeon is a surgeon because the property of being a skilful surgeon must have all

the requisites of the property of being a surgeon, and the additional property of being skilful with respect

to the property of being a surgeon.

• A modiﬁer Mis trivially subsective with respect to Fiﬀ the modiﬁed property [M F ] has exactly the

same requisites as the property F, i.e. if [M F ] and Fshare the same essence, hence are identical

properties. The trivial subsectives are trivial in that the modiﬁcation has no eﬀect on the modiﬁed

property and so might just as well not have taken place.

For instance, there is no semantic or logical (but perhaps rhetorical) diﬀerence between the property of

being a diamond and the property of being a genuine diamond. Trivial modiﬁers such as genuine and real

are pure subsectives: genuine diamonds are not located in the intersection of diamonds and objects that are

genuine, for there is no such property as being genuine, pure and simple. Genuine diamonds form a subset,

though not a proper one, of a given set of diamonds.27

• A modiﬁer Mis privative with respect to Fiﬀ the modiﬁed property [M F ] lacks at least one, but

not all, of the requisites of the property F. Moreover, the essence of [M F ] contains at least one other

requisite that does not belong to the essence of F, and contradicts at least one of the requisites of F.

Hence, Mis privative with respect to Fiﬀ the essence of [M F ] has a non-empty intersection with the

essence of F, and this intersection is a proper subset of both the essences of Fand of [M F ].

25 Contra Kripke, it is not a discovery (a posteriori, yet ‘metaphysically’ necessary) that a domestic cat belongs to any of the

categories above. The deﬁnition of domestic cat in virtue of the conjunction of the above categories is a stipulative deﬁnition,

which is conceptually prior to any empirical discovery of the further properties of various domestic cats (such as weighing seven

pounds, basking on a hot tin roof, or having grey stripes). Our stance is at odds with Kripkean essentialism, as we ﬁnd anyone

conducting empirical inquiry in the animal kingdom needs a conceptual steer on what deserves to be called a domestic cat in the

ﬁrst place before they can claim to have had any sort of causal interaction with domestic cats. (These remarks barely scrape the

surface of a deep philosophical issue, but they serve at least to indicate where we stand.)

26 We are disregarding intersective modiﬁcation in order not to clutter the exposition. However, intersectives are controlled by the

same rule of right subsectivity that applies to the subsectives, together with the special rule of left subsectivity deﬁned in [17].

27 Iwańska [16, p. 350] refers to ‘ideal’, ‘real’, ‘true’, and ‘perfect’ as type-reinforcing adjectives, which seems to get the pragmatics

right of what are semantically pleonastic adjectives. Trivial subsectives should not be confused with subsective intensiﬁers, as in

‘is real pain’, when real pain does not contrast with imaginary pain, but with slight pain.

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For instance, forged banknote has almost the same requisites as does banknote, but it has also another

requisite, namely the property of not being issued by an instance endowed with issuing authority.

To formally deﬁne the diﬀerence between subsective and privative modiﬁcation, we need the TIL deﬁnition

of the relation of being a subset between sets and the operation of the intersection of two sets. The relation

of being a subset between α-sets, ⊆: (o(oα)(oα)), is deﬁned for any type αas follows. Let a, b →v(oα),

x→vα. Then:

0

⊆=λab [∀x[[a x]⊃[b x]]]

The relation of being a proper subset,⊂: (o(oα)(oα)), is then deﬁned as usual:

0

⊂=λab [[∀x[[a x]⊃[b x]]] ∧ ¬[0

=a b]]

For instance, that the set of primes, Prime: (oτ ), is a subset of the naturals, Natural: (oτ ), is captured by

this construction:

[0⊆0

P rime 0

Natural] = [∀x[0

P rime x]⊃[0

N atural x]]

Similarly, that the set of primes is a proper subset of the naturals is captured by this construction:

[0

⊂0

P rime 0

Natural] = [[∀x[0

P rime x]⊃[0

N atural x]] ∧[0

P rime 6=0

Natural]]

The operation of intersection, ∩: ((oα)(oα)(oα)), is deﬁned as follows:

0∩=λab λx [[ax]∧[bx]]

For instance, that the intersection of primes and even numbers, Even: (oτ), is equal to the singleton 2 is

captured by this construction:

[0∩0

P rime 0

Even] = λx [[0

P rime x]∧[0

Ev en x]] = λx [x=02]

In what follows we will use classical (inﬁx) set-theoretical notation for any sets A,B; hence instead of

‘[0

⊂A B]’ we will write ‘[A⊂B]’, and instead of ‘[0∩A B]’ we will write ‘[A∩B]’. Since we will be comparing

sets of properties, the type αis here the type of an individual property, (oι)τω .

We are now able to provide the following two deﬁnitions.

Deﬁnition 6 (Subsective vs. privative modiﬁers). Let M→((oι)τω (oι)τ ω ); F, p →(oι)τω . Then

• A modiﬁer Mis subsective with respect to a property Fiﬀ

[0

Ess F ]⊆[0

Ess [M F ]]

• A modiﬁer Mis non-trivially subsective with respect to a property Fiﬀ

[0

Ess F ]⊂[0

Ess [M F ]]

• A modiﬁer Mis privative with respect to a property Fiﬀ

[[0

Ess F ]∩[0

Ess [M F ]]] 6=∅ ∧

∃p[[[0

Ess F ]p]∧[[0

Ess [M F ]] λwλt [λx ¬[pwtx]]]]

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Remark. The second conjunct deﬁning privative modiﬁer is to be read as follows: “There is a property

psuch that it is a requisite of the property F([[0

Ess F ]p]), and among the requisites of the modiﬁed

property [M F ] there is a property that contradicts p: [[0

Ess [M F ]] λwλt [λx ¬[pwtx]]].” This follows from

the semantics of privative modiﬁcation. The privative modiﬁer Mnot only deprives the property Fof one

or more of its requisites, it also adds at least one requisite that causes privation.

Remark. The above deﬁnition of subsective and privative modiﬁers is a novel contribution of this paper. It

is an improvement over the corresponding deﬁnitions in Primiero and Jespersen [28] and Duží et al. [6, §4.4].

As for subsective modiﬁers, the new deﬁnition diﬀerentiates between non-trivially and trivially subsective

modiﬁers. As for privatives, the original deﬁnition is a logical consequence of this new one, as we are going

to prove below. It not only stipulates that among the requisites of the privatively modiﬁed property Fis

the property of not being an F, but also explains why it is so. Furthermore, the new deﬁnition also speciﬁes

what the modiﬁed property and the root property have in common. Privation deprives the root property

of some but not all of its requisites. The more requisites of the root property Fare preserved, the closer a

relative the modiﬁed property is to F. Thus, we are able to keep track of the root property in the modiﬁed

property, which in turn makes it possible to prove that, for instance, a demolished damaged house is not a

demolished damaged bridge (see below for this example).

Example. The modiﬁer Wooden: ((oι)τ ω (oι)τ ω ) is subsective with respect to the property of being a table,

Table: (oι)τ ω , but privative with respect to the property of being a horse, Horse: (oι)τ ω . Of course, a

wooden table is a table, but the essence of the property [0W ooden 0T able] is enriched by the property of

being wooden. Being wooden is a requisite of the property of being a wooden table, but it is not a requisite

of the property of being a table, because tables can be instead made of stone, iron, glass, etc.

[0

Ess 0

T able]⊂[0

Ess [0W ooden 0T abl e]]

But a wooden horse is not a horse. The modiﬁer Wooden, the same modiﬁer that just modiﬁed Table, deprives

the essence of Horse of many requisites, for instance of the property of being a living thing, or having a

bloodstream, or having kidneys, etc. Hence among the requisites of the property [0W ooden 0Horse] there

are properties like not being a living thing,not having a bloodstream, etc., which are contradictory (not just

contrary) to some of the requisites of the property Horse. On the other hand, the property [0W ooden 0H orse]

shares many requisites with the property of being a horse, like the outline of the body, resemblance of a

horse, etc., and has the additional requisite of being made of wood. Thus, we have (LT : (oι)τ ω , the property

of being a living thing, HB: (oι)τ ω , the property of having blood):

[[0

Ess 0

Horse]∩[0

Ess [0

W ooden 0

Horse]]] 6=∅ ∧

[[[0

Ess 0

Horse]0

LT ]∧[[0

Ess [0

W ooden 0

Horse]] λwλt λx ¬[0

LT x]]] ∨

[[[0

Ess 0

Horse]0

HB]∧[[0

Ess [0

W ooden 0

Horse]] λwλt λx ¬[0

HB x]]] ∨

etc.

At the outset of this paper we characterized the diﬀerence between subsective and privative modiﬁers

by means of the rule of right subsectivity, which holds for subsective but not privative modiﬁers: a skilful

surgeon is a surgeon; a fake banknote fails to be a bank note.

When Ms→((oι)τ ω (oι)τω ) is a construction of a modiﬁer subsective with respect to the property

v-constructed by F→(oι)τω , then necessarily and for all individuals xthe following rule of right subsectivity

(RS) is valid:

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[[MsF]wt x]

[Fwt x]RS

By Deﬁnition 6 it holds that [0Ess F ]⊆[0Ess [MsF]]. Hence each requisite of Fis also a requisite of [MsF],

but not vice versa, provided Msis non-trivially subsective. By Deﬁnition 4 and Claim 1, since each property

is a requisite of itself, it follows that Fis a requisite of [MsF]:

∀w∀t[∀x[[0T ruewt λwλt [[MsF]wt x]] ⊃[0T ruewt λwλt [Fwt x]]]]

which proves the rule of right subsectivity (RS).

For privatives, we already suggested replacing Boolean negation by property negation, denoted by ‘non’,

to specify the rule governing privatives. Let Mp→((oι)τ ω (oι)τ ω ) be a construction of a modiﬁer privative

with respect to the property v-constructed by F→(oι)τω . Then:

[[MpF]wt x]

[[non F ]wt x]Priv

Of course, it also holds that if xis an [MpF] then it is not the case that xis an F:

[[MpF]wt x]

¬[Fwt x]Single Privation

The reason for replacing Boolean negation by property negation is this. For each individual xand for

each property F, it is either true that xis an For it is not true. Yet there are many individuals that

are neither an Fnor a [non F ]. For instance, each individual either is or is not a banknote. Yet most

individuals are neither a banknote nor a fake banknote, because a fake banknote must still have something

in common with a banknote. A well-forged banknote is almost a banknote, because the property of being a

well-forged banknote is a ‘close relative’ of the property of being a banknote, sharing many requisites with

this property. Hence the property [non F ] is not contradictory but only contrary to F. Due to the diﬀerence

between contradictory and contrary properties, the Priv rule is indeterministic between the three forks with

the third fork having a further measure of indeterminacy, whereas the standard rule of single privation is

deterministic. Our strategy being that the non-based rule of privation ought to be extended to all instances

of single privation, the discrepancy between indeterministic and deterministic rules will vanish, as both the

rule of single and the rule of iterated privation will now be indeterministic.

We are now going to deﬁne the property negation non and prove that Priv is valid for privative modiﬁers.

3.2. Property negation

The philosophical source of inspiration is Aristotle’s observations that:

The sentences “It is a not-white log” and “It is not a white log” do not imply one another’s truth. For if

“It is a not-white log” is true, it must be a log: but that which is not a white log need not be a log at

all. (Prior Analytics I, 46, 1.)

That is, in modern parlance, a set of logs divides into those that are white and those that are non-white,

whereas a set of non-(white logs) divides into those elements that are non-white logs and those that are

not even logs (though perhaps white). More speciﬁcally, this quotation has inspired us to adopt property

negation. And directly relevant for our present purpose:

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From the fact that John is not dishonest we cannot conclude that John is honest, but only that he is

possibly so. ([26, p. 255])

The alternative is namely that John is neither dishonest, nor honest, so “John is not dishonest”, if true,

tells us what John fails to be and what the alternatives are: (i) being honest, (ii) neither being honest nor

being dishonest. The contradictory property is that it is not the case that it is not the case that John is

honest, which is logically equivalent to him being honest. More speciﬁcally, this quotation has inspired us to

introduce the trifurcation of cases presented in the Introduction. This trifurcation is epistemic rather than

ontological, as it bears on the (in-)validity of various inferences.

The deﬁnition of property negation must encapsulate the contrariety clause that the intensional negation

of one of two conjuncts that are mutually exclusive does not entail the truth of the other conjunct.

Deﬁnition 7 (Contrary properties). Let x→ι;F, G →(oι)τω . Then the properties F,Gare mutually

contrary iﬀ

∀w∀t∀x[[Fwt x]⊃ ¬[Gwt x]] ∧ ∃w∃t∃x[¬[Fwt x]∧ ¬[Gwt x]]

The deﬁnition states that it is not possible for xto co-instantiate Fand G, and possibly xinstantiates

neither F, nor G. The left-hand conjunct,

∀w∀t∀x[[Fwt x]⊃ ¬[Gwt x]]

is the clause that Fand Gare mutually exclusive. The second conjunct,

∃w∃t∃x[¬[Fwt x]∧ ¬[Gwt x]]

is the contrariety clause that the negation of one of the conjuncts [Fw t x], [Gwt x] does not entail the truth

of the other one.

Next, we want to show that any property [MpF] formed from a property Fby a modiﬁer Mpprivative

with respect to Fis a property contrary to F. First, we prove the left-hand conjunct:

∀w∀t∀x[[[MpF]wt x]⊃ ¬[Fwt x]]

To this end, we apply the second clause of the deﬁnition of privative modiﬁers (Deﬁnition 6): ∃p[[[0

Ess F ]p]∧

[[0

Ess [MpF]] λwλt [λx ¬[pwt x]]]]. Hence the property [MpF] has among its requisites at least one property

contradictory to a requisite of the property F. Let these properties be Pand λwλt [λx ¬[Pwt x]], respectively.

Then at no hw, tiis there an individual xthat would satisfy both [[MpF]wt x] and [Fwt x]; if there were such

an x, then according to Deﬁnitions 4 and 5, xwould also have to satisfy both [Pwt x] and ¬[Pwt x], which

is logically impossible.

Remark. This proves that the previous deﬁnition found in [6, §4.4] and [28] is a corollary of the new

Deﬁnition 6.

The contrariety clause ∃w∃t∃x[¬[[MpF]wt x]∧ ¬[Fwt x]] holds due to the thesis of individual anti-

essentialism which we subscribe to: no individual has any purely contingent property necessarily.

We should not forget, however, the limiting case where Fis a trivial, non-contingent property with a

constant extension, such as being self-identical. In this case, necessarily, when the type is (say) ι,Fwt is the

entire type ιand λ x ¬[Fwt x] is an empty ι-set, because at no hw, tiis there an individual that would be

neither identical with itself nor non-identical with itself. Another example of a non-contingent property is

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the property being identical to a or b.28 At all hw, tithe extension of this property is the set {a, b}, and

at no hw, tiis there an individual that would be neither identical with aor b, nor non-identical with aor

b, for both aand bare necessarily around to instantiate this property. The upshot is that non-contingent

properties do not lend themselves to being modiﬁed by privative modiﬁers on pain of necessary falsehood.

Hence, if Tri is such a trivial non-contingent property then the extension of [MpT ri] is necessarily the

empty ι-set for any modiﬁer privative with respect to Tri. Such a modiﬁer turns Tri into an ‘idle property’

that has necessarily an empty extension.

Deﬁnition 8 (General modiﬁer privative with respect to a property f). Let =: (o((oι)τ ω (oι)τ ω )((oι)τ ω (oι)τ ω ))

be the identity relation deﬁned over ﬁrst-order modiﬁers, non →((oι)τ ω (oι)τ ω ) a variable ranging over

ﬁrst-order modiﬁers, f→(oι)τ ω ,Con : (o(oι)τ ω (oι)τ ω ) the relation of contrariety between properties.

Then:

0

Non =λf λwλt [λx ∃non [[[non f ]wt x]∧[0

Con [non f ]f]]]

is the general modiﬁer privative with respect to f.

Remark. Any of the modiﬁers non meeting the condition speciﬁed by Deﬁnition 8 are privative with respect

to the property F. Property negation takes a particular property Fto an arbitrary property contrary to it,

[non F ].29

Non is thus the unique general privative modiﬁer, and it takes a property Fto the general contrary

property [0N on F ]. For instance, [0Non Banknote] is the general property contrary to the property of being

a banknote. Necessarily, the extension of [0N on Banknote]wt includes the extensions of the properties forged

banknote,banknote dissolved in acid,Monopoly banknote, etc., some of the extensions possibly being empty.

One might worry that it is too much to claim that, necessarily, the extension contains the full panoply of

non-banknotes. But it follows from Deﬁnition 8 that the full panoply is indeed involved. At any hw, ti, for

any individual xand the property constructed by F→(oι)τω , this holds:

[[0

N on F ]wt x] = ∃non [[[non F ]wt x]∧[0

Con [non F ]F]]

Hence, individual ahas the property [0

N on F ] iﬀ ahas any property [non F ] for some non privative with

respect to F. Thus, the set [0

N on F ]wt is almost as large as the complement Fwt of the set Fwt. At some,

but not all, hw, tiit is the case that [0

N on F ]wt =Fwt . Or, when Fwt happens to be the entire type ι,

then [0

N on F ]wt must be the empty ι-set, i.e. the union of all empty sets [0

N on F ]wt. Deﬁnition 8 does not

exclude such modiﬁer functions as do not even have a name in our vernacular.

Contrariety provides the weaker form of negation that is suitable for privative modiﬁers as explained

above. Deﬁnition 8 thus justiﬁes the elimination rule Priv for modiﬁers Mpprivative with respect to property

Fstated above:

[[MpF]wt x]⊢[[0

N on F ]wt x]

The conclusion of Priv states that the predication of Feludes x:Fdoes not get to be predicated of x. For

instance, if the premise is that ais a fake banknote then the conclusion is that ais a Non-banknote, therefore

the property banknote is not predicated of a. Or if bis a wooden horse then bis a Non-horse. But if cis

28 TIL comes with a constant domain. See [6, 378–379].

29 Martin [24, p. 449] says, “Semantically [inﬁnite negation, e.g. non-human] converts a term into one that stands for its non-empty

complement . . . ”. Our prop erty negation do es not come with an ontological restriction such as non-emptiness. However, more

importantly, our ‘non-F’ does not denote a complement set, but a contrary property; what denotes a set is ‘non-Fw t ’.

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awooden bird then it follows neither that cis a Non-horse, nor that cis a horse, because the properties

Non-horse and horse necessarily have still something in common (at least one common requisite), unlike

the properties wooden bird and horse or Non-horse.

Moreover, the partial order deﬁned on sets of requisites makes it possible to compare how close the

privatively modiﬁed property [MpF] is to F. Since [MpF] and Fhave some requisites in common, they are

relatives. For instance, a fake banknote is not a fake passport or even a Non-passport; they are not relatives.

Yet a fake banknote is a close relative of banknotes, closer than, for instance, a Monopoly banknote or a

burnt banknote. From this point of view the most distant relative of the property Fis thus the property

[0

N on F ].

Let us run a test case. Can a paradox be deduced from our theory? Consider this example:

(1) Individual ais a €10 banknote

(2) Whatever is a €10 banknote is a banknote

(3) ais a banknote

(a) ais a forged €100 banknote created by adding a ‘0’ to ‘10’ to form ‘100’

(b) Whatever is a forged banknote is a non-banknote

(c) ais a non-banknote

Contradiction: (3) and (c).

This does not follow, however. One fact is that ais a tampered-with €10 banknote. Yet having a zero

add to ‘10’ does not have to undermine a’s property as a €10 banknote. Hence, amay remain a banknote,

for the modiﬁer €10 is subsective with respect to the property of being a banknote. Another fact is that a

is a forged €100 banknote. From this, however, it does not follow that ais no longer a banknote. It only

follows that ais not a €100 banknote. The property that has been compromised by the attempted forgery

is that of being a €100 banknote, not the property of being a banknote per se. The apparent paradox

arises, because premise (b) fails to state that ais a forged €100 banknote and hence a non-€100 banknote.

Therefore, premise (b) becomes irrelevant. Hence, acan be a €10 banknote (and thus a banknote) while

being a non-€100 banknote.30

3.3. Double privation

We turn next to double privation, which has this form:

[M′

p[MpF]]

Since Mpis privative with respect to [MpF], the intersection of the essences of [M′

p[MpF]] and [MpF]

must be non-empty. And since Mpis privative with respect to F, the intersection of the essences of [MpF]

and Fmust also be non-empty. One may then wonder whether the respective essences of [M′

p[MpF]] and F

can be disjunctive. We think not. There must be an overlap of requisites, and not just of any old properties,

but of carefully chosen ones.

Recall the earthquakes in central Italy in 2016. Many houses, bridges and other buildings and construc-

tions were damaged, some beyond repair. A demolished damaged house is surely not a house, but debris: a

particular object goes through the stages of being a house, then a damaged house and ﬁnally a demolished

damaged house, which is in material terms nothing but debris. Yet a demolished damaged house is diﬀerent

from a demolished damaged bridge. A demolished damaged house shares requisites with houses that it does

not share with demolished damaged bridges.

30 We are indebted to Nikolaj Nottelmann and Lars Binderup for discussion of this example.

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It may so happen that the essence of [M′

p[MpF]] is a superset of the essence of F. In such a case,

if xinstantiates [M′

p[MpF]] then xalso instantiates F. For instance, a repaired damaged house is again

a house. To repair a damage is to undo the damage and in so doing returning the previously damaged

artefact to its still earlier state of functioning properly; such is the semantics of the verb ‘to repair’ and

the adjective ‘repaired’. So here we have come full circle back to F. This particular instance of the modiﬁer

repaired is privative with respect to damaged house, because what is a non-house turns into a house. (We

are presupposing, to get the example oﬀ the ground, that a damaged house is so damaged that it no longer

qualiﬁes as a house.) Being a repaired damaged house is one way of being a house. Formally:

[[0

Ess 0

House]⊂[0

Ess [0

Repaired [0

Damaged 0

House]]]]

Yet it may also so happen that the essence of [M′

p[MpF]] and the essence of Fhave a non-empty

intersection, but neither is a subset of the other. For instance, a demolished damaged house is neither a

damaged house, nor a house, but something altogether diﬀerent, namely a pile of rubble. The modiﬁer

demolished, like repaired above, is privative with respect to damaged house, but the logical eﬀect of applying

it to damaged house is the opposite. The semantics of the verb ‘to demolish’ puts it in opposition to ‘to repair’

or ‘to restore’. Nonetheless, a demolished damaged house must possess the requisite of having previously

been a house.

As is seen, the property of being a demolished damaged house spans three states: ﬁrst, being a house;

second, being a damaged house; third, being a demolished damaged house. Formally:

[0

Ess [0

Demolished [0

Damaged 0

House]]] ∩[0

Ess 0

House]6=∅

Absent the requisite property of having been previously a house, there is nothing to block the inference that

a demolished damaged house is (say) a demolished damaged bridge.

3.4. Three case studies

Here we revisit three examples that were broached above. For better readability of the following formulae,

we will now abbreviate formulae for constructions of the form ‘λwλt [λx ¬[pwt x]]’ as ‘not-p’.

3.4.1. First fork

Since damaged is privative with respect to house, we have (as per Deﬁnition 6):

[[0

Ess 0

House]∩[0

Ess [0

Damaged 0

House]]] 6=∅

∧ ∃p[[[0

Ess 0

House]p]∧[[0

Ess [0

Damaged 0

House]] not-p]]

Hence damaged has turned some of the requisites of house into their opposites. For instance, if one of the

requisites of being a house is the property of being a place to live in, then damaged turns this property into

the property of not being a place to live in. Since repaired is privative with respect to the property damaged

house, we have:

[[0

Ess [0

Damaged 0

House]] ∩[0

Ess [0

Repaired [0

Damaged 0

House]]]] 6=∅

∧ ∃q[[[0

Ess [0

Damaged 0

House]] q]

∧[[0

Ess [0

Repaired [0

Damaged 0

House]]] not-q]]

Now repaired cancels the eﬀect of damaged; it must turn all those opposites not-p of damaged house back

into the original requisites pof House. Thus, among those properties qthat are contained in the essence

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of [0

Damaged 0

House] and appear as not-q in the essence of [0

Repaired [0

Damaged 0

House]] there must be

all those properties pwhich are contained in the essence of house and their opposites not-p in the essence

of [0

Damaged 0

House]. As a result, among these properties qthere are the properties λwλt [λx ¬¬[pwt x]],

hence p. We obtain:

[[0

Ess 0

House]⊂[0

Ess [0

Repaired [0

Damaged 0

House]]]]

The property repaired damaged house has all the requisites of house, being again a place to live in.

3.4.2. Second fork

Contrast the above property with the property demolished damaged house. Whatever is a demolished

damaged house cannot be a house, for the same reason that a demolished house cannot be a house. As soon

as we understand the meaning of the predicate ‘is a demolished damaged house’, we are able to calculate

which way it goes, and that we must land on the second fork. So, we know that a demolished damaged

house is a non-house. But we know something positive about it, too: we know that it is now a pile of rubble.

A demolished damaged house has been physically reduced to its raw matter (wood, steel, brick, etc.), just

like a melted-down statue is reduced to its raw matter (bronze, clay, etc.). The internal link between being

a demolished damaged house and being a pile of rubble is that that pile of rubble has a noble past as a

damaged house and before that as a house.

3.4.3. Third fork

Consider again former heir apparent. This combination of privatives is doubly dynamic due to the

backward-looking aspect of former and the forward-looking aspect of apparent, in the special sense of

‘apparent’ as ‘designated to become’. Someone who is a designated Fis currently not yet an F, though they

are supposed to become one. We are deploying the strict interpretation of ‘former’ as a privative rather

than a modal modiﬁer to get the example of former heir apparent oﬀ the ground.31 With that in place,

someone who is a former heir apparent is not an heir, for one of two reasons: either the person succeeded

in succeeding the previous monarch (promotion), or the person is no longer being even considered for the

throne (demotion).

Accordingly, one requisite which heir apparent comes with is that any bearer must lack the property

of being the successor (where it is understood which is the relevant royal position, e.g. the oﬃce of King

of Denmark): this requisite property is due to the modiﬁcation provided by apparent. Another requisite

which former heir apparent comes with is that any bearer must lack the property of being any longer the

prospective heir. The backward-looking aspect of former voids the forward-looking aspect of apparent, which

brings us to the present time where the bearer of the property of being a former heir apparent may, or may

not, be sitting on the throne.

3.4.4. Summary

To sum up these three case studies, which fork is the right one depends on the semantics of the modiﬁers

involved. When faced with iterated privatives, the agents who operate within some interactive system

for reasoning on the basis of natural-language texts can request additional information about particular

modiﬁers.32 The appropriate answer will be a reﬁnement of the modiﬁer in question. For instance, an

appropriate reﬁnement of repaired would be this:

∀p[[[[0

Ess F ]p]∧[[0

Ess[MpF]] not-p]] ⊃[[0

Ess [0

Repaired [MpF]]] p]]

31 On the privative reading, from “ais a former F” it can be inferred that ais no longer an F, hence is not an F. On the modal

reading, it cannot be excluded that ahas been reinstated as an F.

32 A particular such system is investigated in, e.g., [4] and [10].

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Thus, we can infer that whatever xis a repaired [MpF] is also an F. Similarly, a supplementary piece of

information about the semantics of demolished might be this:

∃p[[[[0

Ess F ]p]∧[[0

Ess[MpF]] not-p]] ∧[[0

Ess [0

Demolished [MpF]]] not-p]]

Then we can infer that a demolished [MpF] is not an F. If no such reﬁnement can be supplied, then we

cannot decide which of the ﬁrst two forks an individual xlands on, and so we know that we are facing a

case of the third fork.

4. Conclusion

The results obtained in this paper amount to an extension of the standard theory of property modiﬁcation

by adding a logic of iterated privation to it. We started out with the problem that the received rule of single

privation is too crude, because it turns the root property into the contradictory property. To start solving

the problem, we replaced Boolean negation by property negation, enabling us to operate on contrary rather

than contradictory properties.

We then assigned so-called requisites to properties, and deﬁned the essence of a property as the set of

all its requisites. Also, properties formed by means of iterated privation are equipped with requisites. They

underpin our presuppositional theory of positive predication, which is the predication of properties an object

must have, as a matter of analytic necessity, if it has a particular privatively modiﬁed property.

The notion of requisite properties enabled us to show that properties formed from iterated privatives give

rise to a trifurcation of cases between returning to the original root property or to a property contrary to

it or being semantically undecidable for want of further information. We have thereby exceeded the general

insight that pairs of privatives yield contraries rather than contradictories, because we are in a position to

calculate which of the forks of the trifurcation we land on.

Acknowledgements

This research has been supported by a University of Padua project on Disagreement, PRAT UNIPD

(Massimiliano Carrara), as well as by the Grant Agency of the Czech Republic Project No. GA15-13277S,

Hyperintensional Logic for Natural Language Analysis, and Project SGS No. SP2017/133 of the internal

grant agency of VŠB-TUO, Knowledge Model ling and its Applications in Software Engineering III (Marie

Duží and Bjørn Jespersen). Various versions of this material have been presented by Bjørn Jespersen as

invited lectures at the Heinrich Heine University Düsseldorf, 21 December 2017; at the Network of Danish

Philosophers Abroad, University of Southern Denmark, 8–9 September 2017; conference How To Say ‘Yes’ or

‘No’: Logical Approaches to Modes of Assertion and Denial, Universitá del Salento, Lecce (21–22 January

2016); Ruhr-Universität Bochum (5 November 2013); Hong Kong University (8 May 2012); Hong Kong

Polytechnic University (7 May 2012); as a tutorial at Technical University of Ostrava (26 February 2013);

and, as solicited lectures at LOGICA 2014, Hejnice (17–20 June 2014) together with Massimiliano Carrara;

CLPS 13, Ghent (16–18 September 2013). This research on iterated privation grew out of a project aimed at

developing an intensional logic for reasoning about technical artefacts in general and technical malfunction

in particular that Massimiliano Carrara and Bjørn Jespersen embarked upon in 2008 when the latter was

aﬃliated with the then-Section of Philosophy, Delft University of Technology. The Section sponsored a

one-month stay in Padua to initiate the project in cooperation with Massimiliano Carrara. Marie Duží later

joined the sub-project on iterated privation. The present paper completes what we would occasionally refer

to as ‘the malfunction trilogy’. We wish to thank two anonymous reviewers for Journal of Applied Logic for

very valuable comments and suggestions which improved the quality of the paper.

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